Power control method for interference alignment in wireless network

ABSTRACT

A power control method for interference alignment in wireless network having K transmitters and K receivers is provided. The method comprising: receiving, performed by receiver n (n is an integer, 1≦n≦K−1), a power indication signal of transmitter n+1 from the transmitter n+1; determining, performed by the receiver n, power of transmitter n; and transmitting, performed by the receiver n, a power indication signal of transmitter n to the transmitter n, wherein the power of transmitter n is determined based on a residual interference of the receiver n, and the power indication signal of transmitter n indicates a minimum transmission power or a maximum transmission power of transmitter n.

TECHNICAL FIELD

The present invention relates to wireless communication, and moreparticularly, to a power control method for interference alignment inwireless network.

BACKGROUND ART

In a wireless network with multiple interfering links, interferencealignment (IA) is used. IA is a transmission scheme achieving linear sumcapacity scaling with the number of data links, at high SNR. With IA,each transmitter designs the precoder to align the interference on thesubspace of allowable interference dimension over the time, frequency orspace dimension, where the dimension of interference at each receiver issmaller than the total number of interferers. Therefore, each receiversimply cancels interferers and acquires interference-free desired signalspace using zero-forcing (ZF) receive filter.

In practical interference alignment (IA) systems, beamformers (precodingvectors) are computed based on limited feedback of channel stateinformation (CSI). Suppressing residual interference caused by CSIinaccuracy can lead to overwhelming network feedback overhead. Analternative solution for coping with residual interference is to controlthe transmission power of all transmitters such that the networkthroughput is maximized. Finding the optimal power control method formulti-antenna IA system requires solving a non-convex optimizationproblem and has no known closed-form solution.

SUMMARY OF INVENTION Technical Problem

It is an object of the present invention to provide a power controlmethod of interference alignment in wireless network nearly optimal.

Solution to Problem

A power control method for interference alignment in wireless networkhaving K transmitters and K receivers, the method comprising: receiving,performed by receiver n (n is an integer, 1≦n≦K−1), a power indicationsignal of transmitter n+1 from the transmitter n+1; determining,performed by the receiver n, power of transmitter n; and transmitting,performed by the receiver n, a power indication signal of transmitter nto the transmitter n, wherein the power of transmitter n is determinedbased on a residual interference of the receiver n, and the powerindication signal of transmitter n indicates a minimum transmissionpower or a maximum transmission power of transmitter n.

Advantageous Effects of Invention

In accordance with the present invention, sum rate capacity of awireless network approaches near optimal result.

BRIEF DESCRIPTION OF DRAWINGS

FIG. 1 shows a framework of the present invention.

FIG. 2 shows a CSI exchange method (method 1).

FIG. 3 illustrates a conventional full-feedback method.

FIG. 4 shows an example of star feedback method (method 3).

FIG. 5 shows the procedure of CSI exchange method in case that K is 4.

FIG. 6 shows the procedure of CSI exchange method based on precoded RSin case that K is 4.

FIG. 7 shows effect of interference misalignment due to quantization offeedback CSI.

FIG. 8 shows probability of turning off for each transmitter.

FIG. 9 shows a binary power control method.

FIG. 10 shows the result of the binary power control method.

FIG. 11 shows a CPC method.

MODE FOR THE INVENTION

In a wireless network with multiple interfering links, interferencealignment (IA) uses precoding to align at each receiver the interferencecomponents from different sources. As a result, in the high SNR regime,the network capacity scales logarithmically with thesignal-to-noise-ratio (SNR) and linearly with half of the number ofparallel subchannels, called degrees of freedom (DOF). In addition, inthe presence of multi-antennas, the capacity also scales linearly withthe spatial DOF per link. IA has been proved to be optimal in terms ofDOF.

This motivates extensive research on IA methods for various types ofchannels and settings, including MIMO channel, cellular networks,distributed IA, IA in the signal space and limited feedback. However,most existing works on IA rely on the impractical assumption that eachtransmitter in an IA network requires perfect CSI of all interferencechannels. Some preliminary results have been obtained on the scalinglaws of numbers of CSI feedback bits with respect to the SNR under theIA constraint. However, there exist no designs of practical CSI feedbackalgorithms for IA networks.

To facilitate the description of present invention, the framework of thepresent invention will be explained firstly.

FIG. 1 shows a framework of the present invention.

Referring FIG. 1, the framework of the present invention comprises acooperative CSI exchange, a transmission power control and ainterference misalignment control.

1. Cooperative CSI exchange: Transmit beamformers will be alignedprogressively by iterative cooperative CSI exchange between interferersand their interfered receivers. In each round of exchange, a subset oftransmitters receive CSI from a subset of receivers and relay the CSI toa different subset of receivers. Based on this approach, CSI exchangemethods are optimized for minimizing the number of CSI transmissionlinks and the dimensionality of exchanged CSI, resulting in smallnetwork overhead.

2. Transmission power control: Given finite-rate CSI exchange,quantization errors in CSI cause interference misalignment. To controlthe resultant residual interference, methods are invented to supportexchange of transmission power control (TPC) signals between interferersand receivers to regulate the transmission power of interferers underdifferent users' quality-of-service (QoS) requirements. Since IA isdecentralized and links are coupled, multiple rounds of TPC exchange maybe required.

3. Interference misalignment control: Besides interferers' transmissionpower, another factor that influences interference power is the degreesof interference misalignment (IM) of transmit beamformers, whichincrease with CSI quantization errors and vice versa. The IM degrees areregulated by adapting the resolutions of feedback CSI to satisfy users'QoS requirements. This requires the employment of hierarchical codebooksat receivers that support variable quantization resolutions. DesigningIM control policies is formulated as an optimization problem ofminimizing network feedback overhead under the link QoS constraints.Furthermore, the policies are optimized using stochastic optimization tocompress feedback in time.

In addition, it will be described a star feedback method, a efficientinterference alignment for 3-cell MIMO interference channel with limitedfeedback and binary power control for IA with limited feedback.

4. Star Feedback Method: For achieving IA, each precoder is aligned toother precoders and thus its computation can be centralized at centralunit, which is called CSI base station (CSI-BS). In star feedbackmethod, CSI-BS gathers CSI of interference from all receivers andcomputes the precoders. Then, each receiver only feeds CSI back toCSI-BS, not to all transmitters. For this reason, star feedback methodcan effectively scale down CSI overhead compared with conventionalfeedback method.

5. Efficient interference alignment for 3 cell MIMO interference channelwith limited feedback.: Conventional IA solution for achieving theoptimal DOF and a special case IA solution requires product of all crosslink channel information since all precoders are coupled. When limitedfeedback channel is assumed, channel matrix product arises a criticalproblem that there is a K-fold increase in error of initial precoder dueto multiplied and summed channel quantization errors. IA solution forachieving the optimal DOF may not be optimal in practical number offeedback bits.

To avoid the product of channel matrices and get better performance, aDOF following method is proposed. If one node does not use a DOF, IAsolution is separated to several independent equation, resulting inavoiding the product of all cross link channel matrices and reducingfeedback overhead.

6. Binary Power Control for Interference Alignment with LimitedFeedback: In practical interference alignment (IA) systems, beamformers(i.e. precoding vector or matrix) are computed based on limited feedbackof channel state information (CSI). Suppressing residual interferencecaused by CSI inaccuracy can lead to overwhelming network feedbackoverhead. An alternative solution for coping with residual interferenceis to control the transmission power of all transmitters such that thenetwork throughput is maximized. Finding the optimal power controlpolicy for multi-antenna IA system requires solving a non-convexoptimization problem and has no known closed-form solution. To overcomethis difficulty, it is proposed the simple binary power control (BPC)method, which turns each the transmission either off or on with themaximum power. Furthermore, we design a distributive algorithm forrealizing BPC. Simulation shows that BPC achieves sum throughput closedto the centralize optimal solution based on an exhaustive search.

Firstly, a system model is described.

I. System Model.

we introduce the system model of K user MIMO interference channel anddefine the metric of network overhead that is required for implementingIA precoder.

Consider K user interference channel, where K transmitter-receiver linksexist on the same spectrum and each transmitter send an independent datastream to its corresponding receiver while it interferers with otherreceivers. Assuming every transmitter-receiver node is equipped with Mantennas, the channel between transmitter and receiver is modeled as M×Mindependent MIMO block channel consisting of path-loss and small-scalefading components. Specifically, it is denoted the channel from the j-thtransmitter and the k-th receiver as d_(kj) ^(−α/2) H^([kj]), where α isthe path-loss exponent, d_(kj) is the distance between transmitter andreceiver and H^([kj]) is M×M matrix of independently and identicallydistributed circularly symmetric complex Gaussian random variables withzero mean and unit variance, denoted as CN(0,1).

Let denote v^([k]) and r^([k]) as a (M×1) beamforming vector and receivefilter at the k-th transceiver, where ∥v^([k])∥²=∥r^([k])∥²=1. Abeamforming vector can be called other terminologies such as beamformeror a precoding vector. Then, the received signal at the k-th receivercan be expressed as below equation.

$\begin{matrix}{y^{\lbrack k\rbrack} = {{H^{\lbrack{kk}\rbrack}v^{\lbrack k\rbrack}s_{k}} + {\sum\limits_{j \neq k}\; {H^{\lbrack{kj}\rbrack}v^{\lbrack j\rbrack}s_{j}}} + {n_{k}.}}} & \left\lbrack {{equation}\mspace{14mu} 1} \right\rbrack\end{matrix}$

and the sum rate is calculated as below equation.

$\begin{matrix}{R_{sum} = {\sum\limits_{k = 1}^{K}\; {\log_{2}\left( {1 + \frac{\frac{P}{d_{kk}^{\alpha}}{{r^{{\lbrack k\rbrack}^{\dagger}}H^{\lbrack{kk}\rbrack}v^{\lbrack k\rbrack}}}^{2}}{{\sum\limits_{k \neq j}\; {\frac{P}{d_{kj}^{\alpha}}{{r^{{\lbrack k\rbrack}^{\dagger}}H^{\lbrack{kj}\rbrack}v^{\lbrack j\rbrack}}}^{2}}} + \sigma^{2}}} \right)}}} & \left\lbrack {{equation}\mspace{14mu} 2} \right\rbrack\end{matrix}$

where s_(k) denotes a data symbol sent by the k-th transmitter withE[|s_(k)|²]=P and n_(k) is additive white Gaussian noise (AWGN) vectorwith covariance matrix σ²I_(M).

Under the assumption of perfect and global CSI, IA aims to aligninterference on the lower dimensional subspace of the received signalspace so that each receiver simply cancels interferers and acquires Kinterference-free signal space using zero-forcing (ZF) receive filtersatisfying following equation.

r ^([k]†) H ^([kj]) v ^([j])=0, ∀k≠j  [equation 3]

Therefore, the achievable sum rate of IA is computed by below equation.

$\begin{matrix}{R_{sum} = {\sum\limits_{k = 1}^{K}\; {\log_{2}\left( {1 + {\frac{P}{\sigma^{2}}d_{kk}^{- \alpha}{{r^{{\lbrack k\rbrack}^{\dagger}}H^{\lbrack{kk}\rbrack}v^{\lbrack k\rbrack}}}^{2}}} \right)}}} & \left\lbrack {{equation}\mspace{14mu} 4} \right\rbrack\end{matrix}$

It is assumed that each receiver, say the m-th receiver, perfectlyestimates all interference channels, namely the set of matrices{H^([mk])}_(k=1) ^(K). Also, it is considered the case where CSI can besent in both directions between a transmitter and a receiver.

II. CSI Feedback Methods.

In MIMO constant channel, the closed form solution of IA in K=3 user ispresented in prior arts. And the solution of K=M+1 with a single datastream at transmitter-receiver pairs is proposed in prior arts. However,such closed form solutions in general K user interference channel arestill open problem. Here, three CSI feedback methods, namely the 1. CSIexchange method (method 1), 2. modified CSI exchange method (method 2)and 3. star feedback method (method 3) are described for achieving IAunder the constraints K=M+1. Also, we compare the efficiency of eachfeedback method with sum overhead defined as the total number of complexCSI coefficients transmitted in the network for a given channelrealization expressed in equation 5.

$\begin{matrix}{N - {\sum\limits_{m,{k \in {\{{1,2,\ldots \mspace{14mu},K}\}}}}\; \left( {N_{TR}^{\lbrack{mk}\rbrack} + N_{RT}^{\lbrack{mk}\rbrack}} \right)}} & \left\lbrack {{equation}\mspace{14mu} 5} \right\rbrack\end{matrix}$

In equation 5, N_(TR) ^([mk]) denotes the integer equal to the number ofcomplex CSI coefficients sent from the k-th receiver to the m-thtransmitter, and N_(RT) ^([mk]) from the k-th transmitter to the m-threceiver.

FIG. 2 shows a CSI exchange method in the case of K=4 (method 1).

Assuming each pair of transmitter-receiver gets one multiplexing gainunder the constraint K=M+1, the set of interference{H^([mk])v^([k])}_(k=1;k≠m) ^(K) at each receiver should be aligned onthe subspace of dimension at most M−1 to satisfy IA condition whichdescribed in equation 3. That is to say, at least two of all interferersare designed on the same subspace at the receiver as below equation.

                                 [equation  6] $\begin{matrix}{{span}\mspace{14mu} \left( {H^{\lbrack{{1K} - 1}\rbrack}v^{\lbrack{K - 1}\rbrack}} \right)} & {= {{span}\mspace{14mu} \left( {H^{\lbrack{1K}\rbrack}v^{\lbrack K\rbrack}} \right)}} & {{at}\mspace{14mu} {receiver}\mspace{14mu} 1} \\{{span}\mspace{14mu} \left( {H^{\lbrack{2K}\rbrack}v^{\lbrack K\rbrack}} \right)} & {= {{span}\mspace{14mu} \left( {H^{\lbrack 21\rbrack}v^{\lbrack 1\rbrack}} \right)}} & {{at}\mspace{14mu} {receiver}\mspace{14mu} 2} \\{{span}\mspace{14mu} \left( {H^{\lbrack 31\rbrack}v^{\lbrack 1\rbrack}} \right)} & {= {{span}\mspace{14mu} \left( {H^{\lbrack 32\rbrack}v^{\lbrack 2\rbrack}} \right)}} & {{at}\mspace{14mu} {receiver}\mspace{14mu} 3} \\\; & \vdots & \; \\{{span}\mspace{14mu} \left( {H^{\lbrack{{KK} - 2}\rbrack}v^{\lbrack{K - 2}\rbrack}} \right)} & {= {{span}\mspace{14mu} \left( {H^{\lbrack{{KK} - 1}\rbrack}v^{\lbrack{K - 1}\rbrack}} \right)}} & {{at}\mspace{14mu} {receiver}\mspace{14mu} K}\end{matrix}$

In equation 6, span(A) denotes the vector space that spanned by thecolumns of A and each equation satisfies that span(H^([k+2k])v^([k]))=span (H^([k+2k+1])v^([k+1])) (i.e.span(v^([k]))=span (H^([k+2k]))⁻¹H^([k+2k+1])v^([k+1]))). Asconcatenating {span (v^([k]))}_(k=1) ^(K), IA beamformers are computedby below equation and then normalized to have unit norm.

                                      [equation  7] $\begin{matrix}\begin{matrix}\begin{matrix}\begin{matrix}{{v^{\lbrack 1\rbrack} = {{any}\mspace{14mu} {eigenvector}\mspace{14mu} {of}\mspace{14mu} \begin{matrix}{\left( H^{\lbrack 21\rbrack} \right)^{- 1}{H^{\lbrack{2K}\rbrack}\left( H^{\lbrack{1K}\rbrack} \right)}^{- 1}H^{\lbrack{{1K} - 1}\rbrack}} \\{\ldots \mspace{14mu} \left( H^{\lbrack 32\rbrack} \right)^{- 1}H^{\lbrack 31\rbrack}}\end{matrix}}}\mspace{14mu}} \\{v^{\lbrack 2\rbrack} = {\left( H^{\lbrack 32\rbrack} \right)^{- 1}H^{\lbrack 31\rbrack}v^{\lbrack 1\rbrack}}}\end{matrix} \\\vdots\end{matrix} \\{v^{\lbrack{K - 1}\rbrack} = {\left( H^{\lbrack{{KK} - 1}\rbrack} \right)^{- 1}H^{\lbrack{{KK} - 2}\rbrack}v^{\lbrack{K - 2}\rbrack}}}\end{matrix} \\{v^{\lbrack K\rbrack} = {\left( H^{\lbrack{1K}\rbrack} \right)^{- 1}H^{\lbrack{{1K} - 1}\rbrack}v^{\lbrack{K - 1}\rbrack}}}\end{matrix}$

The solution of k-th beamformer v^([k]) in equation 7 is seriallydetermined by the product of pre-determined v^([k−1]) and channel matrix(H^([k+1k−1]))⁻¹H^([k+1k−1]).

Hereinafter, using the property that v^([k]) is serially determined, CSIexchange method (method 1) will be described. Hereinafter, it is assumedthat K=4, M=3.

1. First, transmitters determine which interferers are aligned on thesame subspace at each receiver. Then, each transmitter informs itscorresponding receiver of the indices of two interferers to be alignedin the same direction. This process is called set-up.

2. Receiver k(∀k≠3) feeds back the product matrix (H^([21]−1)H^([2K]), .. . , (H^([32]))⁻¹H^([31]) to receiver 3 which determines v^([1]) as anyeigenvector of (H^([21]))⁻¹H^([2K])(H^([1K]))⁻¹H^([1K−1]) . . .(H^([32]))⁻¹H^([31]). Then, v^([1]) is fed back to transmitter 1.

3. Computation of v^([2]), . . . , v^([K−1]): Transmitter k−1 feedsforward v^([k−1]) to receiver k+1. Then, receiver k+1 calculates v^([k])using equation 7 and feeds back v^([k]) to transmitter k. This processis performed for k=2 to K−1.

4. computation of v^([k]).

Transmitter K−1 feeds forward v^([K−1]) to receiver 1. Then, receiver 1calculates v^([K]) using equation 7 and feeds back v^([K]) totransmitter K.

In CSI exchange method, two interferers are aligned at each receiveramong K−1 interferers. Therefore, the receiver requires the indices oftwo interferers to be aligned before starting the computation andexchange procedure of {v^([1]), . . . , v^([K])}. The signals for aset-up are forwarded through a control channel from transmitter tocorresponding receiver. These set-up signals are forwarded throughB_(setup) bits control channel from transmitter to correspondingreceiver and its overhead is computed in following lemma.

Lemma 1. The set-up overhead for the CSI exchange method is given asbelow equation.

$\begin{matrix}{B_{setup} = \left\lceil {K \cdot {\log_{2}\left( \frac{\left( {K - 1} \right)\left( {K - 2} \right)}{2} \right)}} \right\rceil} & \left\lbrack {{equation}\mspace{14mu} 8} \right\rbrack\end{matrix}$

Each receiver has K−1 interferers from other transmitters. Therefore,_(K−1)C₂ groups of aligned interferers exist at each receiver, where_(K−1)C₂=(K−1)(K−2)/2. To inform each of K receivers about the group ofaligned interferers, total

$\left\lceil {K \cdot {\log_{2}\left( \frac{\left( {K - 1} \right)\left( {K - 2} \right)}{2} \right)}} \right\rceil$

bits are required for the set-up signaling.

Once the aligned interferers are indicated at the receiver, eachreceiver feeds back CSI of the selected interferers in given channelrealization and the beamformers v^([1]), v^([2]), . . . , V^([K]) aresequentially determined by method 1. The overhead for the CSI exchangemethod can be measured in terms of the number of exchanged complexchannel coefficients. Such overhead is specified in the followingproposition.

Proposition 1.

For the CSI exchange method in MIMO channel, the network overhead isgiven as below equation.

N _(EX)−(K−1)M ²+(2K−1)M  [equation 9]

All receivers inform CSI of the product channel matrices to receiver 3,which comprises (K−1)M² nonzero coefficients. After computing thebeamformer v^([1]), M nonzero coefficients are required to feed it backto transmitter 1. Each beamformer is determined by iterative precoderexchange between transmitters and interfered receivers. In each round ofexchange, the number of nonzero coefficients of feedforward and feedbackbecomes 2M. Thus, total network overhead comprises (K−1)M²+(2K−1)M.

FIG. 3 illustrates a conventional full-feedback method.

For comparison, the sum overhead for the convention CSI feedback methodis illustrated. In the existing IA literature, the design of feedbackmethod is not explicitly addressed. Existing works commonly assume CSIfeedback from each receiver to all its interferers, corresponding to thefull-feedback method illustrated in FIG. 3. For such a conventionalmethod, each receiver transmits the CSI {H^([mk])}_(k=1) ^(K) to each ofits K−1 transmitters and the resultant sum overhead is given in thefollowing lemma.

Lemma 2. The sum overhead of the full-feedback method is given as belowequation.

N _(FF) −K ²(K−1)M ²  [equation 10]

Each receiver feeds back K×M² nonzero coefficients to K−1 interferers.Since K receivers feed back CSI, total network overhead comprisesK²(K−1)M².

From the above result, the overhead N_(FF) increases approximately asK³M² whereas the network throughput grows linearly with K. Thus thenetwork overhead is potentially a limiting factor of the networkthroughput. By comparing proposition 1 and lemma 2, with respect to thefull-feedback method, the CSI exchange method requires much less sumoverhead for achieving IA, namely on the order of KM².

In the CSI exchange method described above, v^([1]) is solved by theeigenvalue problem that incorporates the channel matrices of allinterfering links. However, it still requires a huge overhead innetworks with many links or antennas. To suppress CSI overhead ofv^([1]), we may apply the follow additional constraints indicated by abelow equation.

span(H ^([i1]) v ^([1]))=span(H ^([i2]) v ^([2]))

span(H ^([i+11]) v ^([1]))=span(H ^([i+12]) v ^([2]))  [equation 11]

In equation 11, i is not 1 or 2. As v^([1]) and v^([2]) are aligned onsame dimension at receiver i and i+1, v^([1]) is computed as theeigenvalue problem of (H^([i1]))⁻¹H^([i2])(H^([i+12]))⁻¹H^([i+11]) thatconsists of two interfering product matrices.

Applying the properties expressed in equation 11 at receiver K−1 and K,it can be reformulated IA condition under K=M+1 constraints as belowequation.

                                [equation  12] $\begin{matrix}{{span}\mspace{14mu} \left( {H^{\lbrack 12\rbrack}v^{\lbrack 2\rbrack}} \right)} & {= {{span}\mspace{14mu} \left( {H^{\lbrack 13\rbrack}v^{\lbrack 3\rbrack}} \right)}} & {{at}\mspace{14mu} {receiver}\mspace{14mu} 1} \\{{span}\mspace{14mu} \left( {H^{\lbrack 23\rbrack}v^{\lbrack 3\rbrack}} \right)} & {= {{span}\mspace{14mu} \left( {H^{\lbrack 24\rbrack}v^{\lbrack 4\rbrack}} \right)}} & {{at}\mspace{14mu} {receiver}\mspace{14mu} 2} \\\; & \vdots & \; \\{{span}\mspace{14mu} \left( {H^{\lbrack{K - 11}\rbrack}v^{\lbrack 1\rbrack}} \right)} & {= {{span}\mspace{14mu} \left( {H^{\lbrack{K - 12}\rbrack}v^{\lbrack 2\rbrack}} \right)}} & {{{at}\mspace{14mu} {receiver}\mspace{14mu} K} - 1} \\{{span}\mspace{14mu} \left( {H^{\lbrack{K\; 1}\rbrack}v^{\lbrack 1\rbrack}} \right)} & {= {{span}\mspace{14mu} \left( {H^{\lbrack{K\; 2}\rbrack}v^{\lbrack 2\rbrack}} \right)}} & {{at}\mspace{14mu} {receiver}\mspace{14mu} K}\end{matrix}$

It follows that v^([1]), v^([2]), . . . , v^([K]) can be selected asbelow equation.

                                [equation  13] $\begin{matrix}\begin{matrix}\begin{matrix}\begin{matrix}{v^{\lbrack 1\rbrack} = {{eigenvector}\mspace{14mu} {of}\mspace{14mu} \left( H^{\lbrack{K - 11}\rbrack} \right)^{- 1}{H^{\lbrack{K - 12}\rbrack}\left( H^{\lbrack{K\; 2}\rbrack} \right)}^{- 1}H^{\lbrack{K\; 1}\rbrack}}} \\{v^{\lbrack 2\rbrack} = {\left( H^{\lbrack{K\; 2}\rbrack} \right)^{- 1}H^{\lbrack{K\; 1}\rbrack}v^{\lbrack 1\rbrack}}}\end{matrix} \\{v^{\lbrack 3\rbrack} = {\left( H^{\lbrack 13\rbrack} \right)^{- 1}H^{\lbrack 12\rbrack}v^{\lbrack 2\rbrack}}}\end{matrix} \\\vdots\end{matrix} \\{v^{\lbrack K\rbrack} = {\left( H^{\lbrack{K - {2K}}\rbrack} \right)^{- 1}H^{\lbrack{K - {2\mspace{11mu} K} - 1}\rbrack}v^{\lbrack{K - 1}\rbrack}}}\end{matrix}.$

Using the equation 13, the CSI exchange method can be modified. Themodified CSI exchange method can be called a method 2 for a convenience.The method 2 comprises following procedures.

1. First, transmitters determine which interferers are aligned on thesame subspace at each receiver. Then, each transmitter informs itscorresponding receiver of the indices of two interferers to be alignedin the same direction (set-up).

2. Receiver K forwards the matrix (H^([K2]))⁻¹H^([K1]) to receiver K−1.Then, receiver K−1 computes v^([1]) as any eigenvector of(H^([K−11]))⁻¹H^([K−12])(H^([K2]))⁻¹H^([K1]) and feeds back it to thetransmitter 1.

3. Transmitter 1 feeds forward v^([1]) to receiver K. Then, receiver Kcalculates v^([2]) and feeds back v^([2]) to transmitter 2.

4. Computation of v^([3]), . . . , v^([K])

Transmitter k−1 feeds forward v^([k−1]) to receiver k−2. Then, receiverk−2 calculates v^([k]) and feeds back v^([1]) to transmitter k−1. Thisprocess is performed for k=3 to K.

The corresponding sum overhead of modified CSI exchange method (method2) is given in the following equation.

N _(MEX) −M ²+(2K−1)M  [equation 14]

Comparing the method 2 with the method 1, both methods show the sameburden of overhead for the exchange of beamformers. However, the productchannel matrix for v I′J in Modified CSI exchange method requiresconstant M² overhead in any K user case while overhead of v^([1]) in CSIexchange method increases with KM².

The CSI exchange methods (method 1, method 2) in previous descriptiondegrade the amount of network overhead compared with conventional fullfeedback method. However, it requires 2K−1 iterations for the exchangeof beamformers in method 1 and 2. As the number of iterations isincreased, a full DoF in K user channel can not be achievable since itcauses time delay that results in significant interference misalignmentfor fast fading. To compensate for these drawbacks, we suggest the starfeedback method illustrated in FIG. 4.

FIG. 4 shows an example of star feedback method (method 3).

Referring FIG. 4, wireless network comprises a CSI-base station, aplurality of transmitters, a plurality of receivers. The wirelessnetwork using the star feedback method comprises an agent, called theCSI base station (CSI-BS) which collects CSI from all receivers,computes all beamformers using IA condition in equation 7 or equation 13and sends them back to corresponding transmitters. This method 3 isfeasible since the computation of beamformers for IA is linked with eachother and same interference matrices are commonly required for thosebeamformers. In large K, the star method allows much smaller delaycompared with CSI exchange methods (method 1, method 2).

Star feedback method for IA in wireless network will be described.

1. Initialization: CSI-BS determines which interferers are aligned onthe same dimension at each receiver. Then, it informs each receiver ofthe required channel information for computing IA.

2. Computation of v^([1]), . . . , v^([K]): The CSI-BS collects CSI fromall receivers that comprises K×M² nonzero coefficients and computesbeamformers v^([1]), . . . , v^([K]) with the collected set of CSI.

3. Broadcasting v^([1]), . . . , v^([K]): CSI-BS forwards v^([k]) totransmitter k for k=1, . . . , K, which requires M nonzero coefficientto each of K transmitters.

For the star feedback method in MIMO channel, the network overhead isgiven as below equation.

N _(SF)=(M ² +M)K  [equation 15]

Star feedback method requires only two time slots for computation of IAbeamformers in any number of user K. Therefore, it is robust againstchannel variations due to the fast fading while CSI exchange methods areaffected by 2K−1 slot delay for implementation. However, the networkoverhead of star feedback method is increased with KM² which is largerthan 2KM in modified CSI exchange method. Also, star feedback methodrequires CSI-BS that connects all pairs of transmitter-receiver shouldbe implemented as the additional costs.

The Network overhead of star feedback method can be quantified as below.

i) The overhead for initialization: Each receiver has K−1 interferersfrom other transmitters. Therefore, _(k−1)C₂ groups of alignedinterferers exist at each receiver, where _(k−1)C₂=(K−1)(K−2)/2. Toinform each receiver about the group of aligned interferers, total Klog₂((K−1)(K−2)/2) bits are required for initial signaling.

ii) Feedback overhead for star feedback method: The CSI-BS collects CSIof product channels from all receivers that comprises KM² nonzerocoefficients. After computing the precoding vectors, CSI-BS transmits aprecoder of M nonzero coefficient to each of K transmitters. Therefore,(KM²+2M) nonzero coefficients are required for feedback in star feedbackmethod.

The above result shows that the network overhead for the star feedbackmethod is a linear function of K in contrast with the cubic function forconventional feedback method in MIMO channel. Thus the former methodleads to a much slower increase of network overhead with K.

III. Effect of CSI Feedback Quantization

In the precoding description, the CSI feedback methods are designed onthe assumption of perfect CSI exchange. However, CSI feedback fromreceiver to transmitter requires the channel quantization under thefinite-rate feedback constraints in practical implementation. Thisquantized CSI causes the degradation of system performance due to theresidual interference at each receiver. In this section, we characterizethe throughput loss as the performance degradation in limited feedback.Furthermore, we derive an upper-bound of sum residual interference asthe throughput loss and analyze it as a function of the number offeedback bits in given channel realization.

A. Throughput Loss in Limited Feedback

For the analytical simplicity, let define the throughput loss, ΔR_(sum)as below equation.

$\begin{matrix}\begin{matrix}{{\Delta \; R_{sum}}:={E_{H}\begin{bmatrix}{{\sum\limits_{k = 1}^{K}{\log_{2}\left( \frac{{Pd}_{kk}^{- a}{{r^{{\lbrack k\rbrack}1}H^{\lbrack{kk}\rbrack}v^{\lbrack k\rbrack}}}^{2}}{\sigma^{2}} \right)}} -} \\{\sum\limits_{k = 1}^{K}{\log_{2}\left( \frac{{Pd}_{kk}^{- a}{{r^{{\lbrack k\rbrack}1}H^{\lbrack{kk}\rbrack}v^{\lbrack k\rbrack}}}^{2}}{{\hat{I}}^{\lbrack k\rbrack} + \sigma^{2}} \right)}}\end{bmatrix}}} \\{= {E_{H}\left\lbrack {\sum\limits_{k = 1}^{K}{\log_{2}\left( {\hat{I}}^{\lbrack k\rbrack} \right)}} \right\rbrack}} \\{\overset{(a)}{\leq}{E_{H}\left\lbrack {K\; {\log_{2}\left( {\frac{1}{K}{\sum\limits_{k = 1}^{K}{\hat{I}}^{\lbrack k\rbrack}}} \right)}} \right\rbrack}}\end{matrix} & \left\lbrack {{equation}\mspace{14mu} 16} \right\rbrack\end{matrix}$

In equation 16, {circumflex over (v)}^([k]) and {circumflex over(r)}^([k])

are the transmit beamformer and receive filter based on the quantizedCSI and ‘(a)’ follows from the characteristic of concave functionlog(x), denoting

${\hat{I}}^{\lbrack k\rbrack} - {\sum\limits_{{j = 1},{j \neq k}}^{K}{\frac{R}{d_{kj}^{a}}{{{\hat{r}}^{{\lbrack k\rbrack}1}H^{\lbrack{kj}\rbrack}{\hat{v}}^{\lbrack j\rbrack}}}^{2}}}$

Applying Jensen's inequality, the throughout loss is upper-bounded bybelow equation.

$\begin{matrix}{{\Delta \; R_{sum}} \leq {K\; {\log_{2}\left( {\frac{1}{K}{E_{H}\left\lbrack {\sum\limits_{k = 1}^{K}{\hat{I}}^{\lbrack k\rbrack}} \right\rbrack}} \right)}}} & \left\lbrack {{equation}\mspace{14mu} 17} \right\rbrack\end{matrix}$

In equation 17, ΔR_(sum) is significantly affected by the sum residualinterference

${\hat{I}}^{sum} = {\sum\limits_{k = 1}^{K}{{\hat{I}}^{\lbrack k\rbrack}.}}$

Therefore, the minimization of sum rate loss is equivalent to theminimization of sum residual interference caused by limited feedback bitconstraints.

B. Residual Interference in the Proposed CSI Feedback Methods

Prior to deriving the bound of sum residual interference in proposed CSIfeedback methods, we quantify the quantization error with RVQ using thedistortion measure. Let denote h:=vec(H)/∥H∥, ĥ:−vec(Ĥ)/∥Ĥ∥_(F) and thephase rotation, e^(jφ):=ĥ^(t)h/|ĥ^(t)h|, where H, ĤεC^(N×M).

Then, h can be expressed by below equation.

h=e ^(jφ) ĥ+Δh  [equation 18]

In equation 18, Δh represents the difference between h and e^(jφ)ĥ. H isfollows below equation. ΔH is defined such that vec(ΔH)−Δh.

H=e ^(jφ) Ĥ+ΔH  [equation 19]

Lemma 3. The expected squared norm of the random matrix Δh is bounded asbelow equation.

$\begin{matrix}{{E\left\lbrack {{\Delta \; H}}_{F}^{2} \right\rbrack} \leq {2\; {{\overset{\_}{\Gamma}({MN})} \cdot 2^{- \frac{B}{{MN} - 1}}}}} & \left\lbrack {{equation}\mspace{14mu} 20} \right\rbrack\end{matrix}$

In equation 20, B denotes quantization bit for H and

${\overset{\_}{\Gamma}({MN})} = \frac{2\; {\Gamma \left( \frac{1}{{MN} - 1} \right)}}{{MN} - 1}$

Using properties of lemma 3, we rewrite equation 18 and equation 19 asbelow equation.

h−e ^(jφ) ĥ+σ _(Δh) Δ{tilde over (h)}

H=e ^(jφ) Ĥ+σ _(Δh) Δ{tilde over (H)},  [equation 21]

In equation 21,

${{\Delta \; h} = {\sigma_{\Delta \; h}\Delta \; \overset{\_}{h}}},{{E\left\lbrack {{\Delta \; \overset{\_}{h}}}^{2} \right\rbrack} = 1},{{E\left\lbrack {{\Delta \; \overset{\_}{H}}}_{F}^{2} \right\rbrack} = {1\mspace{14mu} {and}}}$$\sigma_{\Delta \; h}^{2} \leq {{\overset{\_}{\Gamma}({MN})} \cdot {2^{- \frac{B}{{MN} - 1}}.}}$

Also, we derive the error bound of eigenvector of the quantized matrix,which is applied to analyze the sensitivity towards quantization erroron the initialization of v^([1]). Based on the modeling of quantizationerror of H in equation 22, we derive the eigenvector of Ĥ using theperturbation theory in following lemma.

Lemma 4. For a large B, the m-th eigenvector {circumflex over (v)}_(m)of Ĥ is given as below equation.

$\begin{matrix}{{\hat{v}}_{m} = {^{{- j}\; \phi}\left( {v_{m} - {\sigma_{\Delta \; h}{\sum\limits_{{k = 1},{k \neq m}}^{M}{\frac{v_{k}^{*}\Delta \; \overset{\sim}{H}\; v_{m}}{\lambda_{m} - \lambda_{k}}v_{k}}}}} \right)}} & \left\lbrack {{equation}\mspace{14mu} 22} \right\rbrack\end{matrix}$

In equation 22, v_(m) and λ_(m) are the m-th eigenvector and eigenvalueof H.

1) Modified CSI exchange method: As the modified CSI exchange methodrequires smaller sum overhead than that of CSI exchange method, wederive the upper bound of residual interference in modified CSI exchangemethod that consists of two types of feedback channel links: i) Feedbackof the channel matrix for the initial v^([1]) and Sequential exchange ofthe quantized beamformer between transmitter and receiver.

For the initialization of v^([1]), the K-th receiver transmits theproduct channel matrix H_(eff) ^([K]) to the (K−1)^(th) receiver for thecomputation of v^([1]) which is designed as the eigenvector of H_(eff)^([K−1])H_(eff) ^([K]), where

$H_{eff}^{\lbrack{K - 1}\rbrack}:={{\frac{\left( H^{\lbrack{K - 11}\rbrack} \right)^{- 1}H^{\lbrack{K - 12}\rbrack}}{{{\left( H^{\lbrack{K - 11}\rbrack} \right)^{- 1}H^{\lbrack{K - 12}\rbrack}}}_{F}}\mspace{14mu} {and}\mspace{14mu} H_{eff}^{\lbrack K\rbrack}}:=\frac{\left( H^{\lbrack{K2}\rbrack} \right)^{- 1}H^{\lbrack{K1}\rbrack}}{{{\left( H^{\lbrack{K2}\rbrack} \right)^{- 1}H^{\lbrack{K1}\rbrack}}}_{F}}}$

Assuming that RVQ is applied and the quantized matrix Ĥ_(eff) ^([K]) istransmitted through B_(initial) bits feedback channel, the quantizedchannel matrix is expressed as below equation.

$\begin{matrix}{{\hat{H}}_{eff}^{\lbrack K\rbrack} - {^{{- j}\; \phi_{initial}}\left( {H_{eff}^{\lbrack K\rbrack} - {\sigma_{initial}\Delta \; {\hat{H}}_{eff}^{\lbrack K\rbrack}}} \right)}} & \left\lbrack {{equation}\mspace{11mu} 23} \right\rbrack\end{matrix}$

In equation 23,

$\sigma_{initial}^{2} \leq {{\overset{\_}{\Gamma}\left( M^{2} \right)}2^{- B_{\underset{M^{2} - 1}{initial}}}}$and E[Δ Ĥ_(eff)^([K])_(F)²] = 1.

Then, K−1-th receiver computes the quantized beamformer {circumflex over(v)}^([1]) of H_(eff) ^([K−1])Ĥ_(eff) ^([K]) as below equation.

{circumflex over (v)} ^([1]) −e ^(−jφ) ^(initial)   [equation 24]

In equation 24, the quantization error Δv^([1]) is expressed by belowequation.

$\begin{matrix}{{\Delta \; v^{\lbrack 1\rbrack}} = {\sigma_{initial}{\sum\limits_{{k = 1},{k \neq m}}^{M}{\frac{v_{k}^{*}H_{eff}^{\lbrack{K - 1}\rbrack}\Delta \; {\hat{H}}_{eff}^{\lbrack K\rbrack}v_{m}}{\lambda_{m} - \lambda_{k}}v_{k}}}}} & \left\lbrack {{equation}\mspace{14mu} 25} \right\rbrack\end{matrix}$

In equation 25, v_(m) and λ_(i) are the m-th eigenvector and eigenvalueof H_(eff) ^([K−1])H_(eff) ^([K]).

Since the magnitude of quantization error ∥Δv^([1])∥² is represented asbelow equation, the quantization error of limited feedback is affectedby exponential function of B_(initial) and the distribution ofeigenvalues |λ_(m)−λ_(k)|.

$\begin{matrix}{{{\Delta \; v^{\lbrack 1\rbrack}}}^{2} \propto {\sum\limits_{{k = 1},{k \neq m}}^{M}\frac{\sigma_{initial}^{2}}{{{\lambda_{m} - \lambda_{k}}}^{2}}}} & \left\lbrack {{equation}\mspace{14mu} 26} \right\rbrack\end{matrix}$

Secondly, the effect of quantization error due to the exchange ofbeamformers between transmitters and receivers is described. On theassumption of perfect CSI of H_(eff) ^([K]) at receiver K−1, it isconsidered the limited feedback links from receiver to transmitter,where B_(k) bits are allocated to quantize {tilde over (v)}^([k]).

From modified CSI exchange method (method 2), receiver K−1 computesv^([1]) and quantize it to {circumflex over (v)}^([1]) with RVQ andfeeds back to transmitter 1 through B₁ feedback channel. The quantizedbeamformer {circumflex over (v)}^([1]) is represented as below equation.

{circumflex over (v)} ^([1]) =e ^(−jφ) ^(K−1) (v ^([1])−σ₁ Δv^([1]))  [equation 27]

In equation 27,

$\sigma_{1}^{2} = {{\Gamma (M)}2^{- \frac{B_{1}}{M - 1}}}$ andE[Δ v^([1])²] = 1.

Following the procedure of method 2, transmitter 1 forwards {circumflexover (v)}^([1]) to receiver K. The receiver K designs {dot over(v)}^([2]) on the subspace of H_(eff) ^([K]){circumflex over (v)}^([1])as below equation.

$\begin{matrix}\begin{matrix}{{\overset{.}{v}}^{\lbrack 2\rbrack} = {H_{eff}^{\lbrack K\rbrack}{\hat{v}}^{\lbrack 1\rbrack}}} \\{= {^{{- j}\; \phi_{1}}\left( {{H_{eff}^{\lbrack K\rbrack}v^{\lbrack 1\rbrack}} - {\sigma_{1}H_{eff}^{\lbrack K\rbrack}\Delta \; v^{\lbrack 1\rbrack}}} \right)}} \\{= {^{{- j}\; \phi_{1}}\left( {v^{\lbrack 2\rbrack} - {\sigma_{1}H_{eff}^{\lbrack K\rbrack}\Delta \; v^{\lbrack 1\rbrack}}} \right)}}\end{matrix} & \left\lbrack {{equation}\mspace{14mu} 28} \right\rbrack\end{matrix}$

Then, receiver K quantizes {dot over (v)}^([2]) is quantized to{circumflex over (v)}^([2]) with B₂ bit quantization level and informedto transmitter 2. The quantized precoder {circumflex over (v)}^([2]) isrepresented as below equation.

{circumflex over (v)} ^([2]) =e ^(−jφ) ² ({dot over (v)} ^([2])−σ₂ Δv^([2]))  [equation 29]

In equation 29, σ₂ ²= Γ(M)2⁻ ^(M1) ^(B2) and E[∥Δv^([2])∥²]−1.

Likewise, the other beamformers {dot over (v)}^([3]), {dot over(v)}^([4]), . . . , and {dot over (v)}^([K]) are designed at receiver 1,2, . . . , and K−2 and their quantized beamformers {circumflex over(v)}^([3]), {circumflex over (v)}^([4]), . . . , and {circumflex over(v)}^([K]) are fed back to their corresponding transmitters, where{circumflex over (v)}^([k]) is modeled as below equation.

{circumflex over (v)} ^([k]) =e ^(−jφ) ^(k) ({dot over (v)} ^([k])−σ_(k)Δv ^([k]))  [equation 30]

In equation 30,

$\sigma_{k}^{2} = 2^{- \frac{B_{k}}{M - 1}}$ andE[Δ v^([k])²] = 1, for  k = 3, 4, …  , K.

Using equation 28, 29 and 30, the sum residual interference affected byquantization error can be analyzed. To analyze the residual interferenceat each receiver, it is assumed that each receiver designs azero-forcing receiver with a full knowledge of {{circumflex over(v)}^([k]): 1≦k≦K}, which cancels M−1 dimensional interferers. Then, theupper-bound of expected sum residual interference is expressed as afunction of the feedback bits, the eigenvalue of fading channel anddistance between the pairs of transmitter-receiver is suggested asbelow.

Proposition 3. In modified CSI feedback method, the upper-bound ofexpected residual interference at each receiver is represented as belowequation in given feedback bits {B_(k)}_(k−1) ^(K) and channelrealization {H^([jk])}_(j,k=1) ^(K).

$\begin{matrix}\left\{ \begin{matrix}{{{E\left\lbrack {\hat{I}}^{\lbrack k\rbrack} \right\rbrack} \leq {{Pd}_{{kk} + 2}^{- \alpha}\sigma_{k|2}^{2}\lambda_{{ma}\; x}^{\lbrack{{kk} + 2}\rbrack}}},{{{for}\mspace{14mu} k} = 1},\ldots \mspace{14mu},{K - 2}} \\{{E\left\lbrack {\hat{I}}^{\lbrack{K - 1}\rbrack} \right\rbrack} \leq {{Pd}_{K - 11}^{- \alpha}\left( {{\sigma_{2}^{2}\lambda_{{ma}\; x}^{\lbrack{K - 12}\rbrack}} + {\sigma_{1}^{2}\lambda_{{ma}\; x}^{\lbrack{K - 12}\rbrack}} + {\sigma_{1}^{2}\lambda_{{ma}\; x}^{\lbrack{K - 11}\rbrack}}} \right)}} \\{{E\left\lbrack {\hat{I}}^{\lbrack K\rbrack} \right\rbrack} \leq {{Pd}_{K\; 2}^{- \alpha}\sigma_{2}^{2}\lambda_{{ma}\; x}^{\lbrack{K\; 2}\rbrack}}}\end{matrix} \right. & \left\lbrack {{equation}\mspace{14mu} 31} \right\rbrack\end{matrix}$

2) Star feedback method: In star feedback method, it is assumed that allreceivers are connected to CSI-BS with high capacity backhaul links,such as cooperative multicell networks. Then, CSI-BS is allowed toacquire full knowledge of CSI estimated at each receiver. Based onequation 13, CSI-BS computes v^([1]), v^([2]), . . . , v^([K]) andforwards them to the corresponding transmitters. In this section, weconsider the B_(k) feedback bits constraint from CSI-BS to transmitter kand VII is quantized as below equation.

{circumflex over (v)} ^([k]) =e ^(−jφ) ^(k) (v ^([k])−σ_(k) Δv^([k]))  [equation 32]

In equation 32,

$\sigma_{k}^{2} = {{\overset{\_}{\Gamma}(M)}2^{- \frac{B_{k}}{M - 1}}}$and E[Δ v^(p[k])²] − 1.

Given the quantized beamformer, each transmitter sends a data streamthat causes the residual interference at the receiver. Followingproposition provides the upper-bound of expected residual interferenceat each receiver that consists of the feedback bits, the eigenvalue offading channel and distance between the pairs of transmitter-receiver.

Proposition 4. In star feedback method, the upper-bound of expectedresidual interference at each receiver is represented in given feedbackbits {B_(k)}_(k=1) ^(K) and channel realization {H^([jk]}) _(j,k=1) ^(K)as below equation.

$\begin{matrix}\left\{ {{{\begin{matrix}{{E\left\lbrack {\hat{I}}^{\lbrack k\rbrack} \right\rbrack} \leq {{{Pd}_{{kk} + 2}^{- \alpha}\sigma_{k + 1}^{2}\lambda_{}^{\lbrack{{kk} + 1}\rbrack}} + {{Pd}_{{kk} + 2}^{- \alpha}\alpha_{k + 2}^{2}\lambda_{{ma}\; x}^{\lbrack{{kk} + 2}\rbrack}}}} \\{{E\left\lbrack {\hat{I}}^{\lbrack{K - 1}\rbrack} \right\rbrack} \leq {{{Pd}_{K - 11}^{- \alpha}\sigma_{1}^{2}\lambda_{{ma}\; x}^{\lbrack{K - 11}\rbrack}} + {{Pd}_{K - 11}^{- \alpha}\sigma_{2}^{2}\lambda_{{ma}\; x}^{\lbrack{K - 2}\rbrack}}}} \\{{E\left\lbrack {\hat{I}}^{\lbrack K\rbrack} \right\rbrack} \leq {{{Pd}_{K\; 2}^{- \alpha}\sigma_{1}^{2}\lambda_{{ma}\; x}^{\lbrack K\rbrack}} + {{Pd}_{K\; 2}^{- \alpha}\alpha_{2}^{2}{\lambda_{{ma}\; x}^{\lbrack{K\; 2}\rbrack}.}}}}\end{matrix}{for}\mspace{14mu} k} = 1},\ldots \mspace{14mu},{K - 2.}} \right. & \left\lbrack {{equation}\mspace{14mu} 33} \right\rbrack\end{matrix}$

IV. Dynamic Feedback Bit Allocation Methods for CSI Feedback Methods

The system performance of IA is significantly affected by the number offeedback bits that induces residual interference in a finite ratefeedback channel. The dynamic feedback bit allocation strategies thatminimize the throughput loss due to the quantization error in given sumfeedback bits constraint is explained. Furthermore, It is provided therequired number of feedback bits achieving a full network DoF in eachCSI feedback methods.

A. Dynamic Bit Allocation for Minimizing Throughput Loss

Consider total B_(tot) bits are given for the feedback framework in thenetwork model. Since the throughput loss is bounded by the expected sumresidual interference, we can formulate the dynamic bit allocationproblem for minimizing throughput loss as follows equation.

$\begin{matrix}{{\min \; {\sum\limits_{k - 1}^{K}{E\left\lbrack {\hat{I}}^{\lbrack k\rbrack} \right\rbrack}}}{{s.t.\mspace{14mu} {\sum\limits_{k = 1}^{K}B_{k}}} \leq B_{tot}}} & \left\lbrack {{equation}\mspace{14mu} 34} \right\rbrack\end{matrix}$

Using proposition 3 and proposition 4, equation 34 can be transformed toconvex optimization problem with variables {B_(k)}_(k=1) ^(K) expressedby below equation.

$\begin{matrix}{{\min {\sum\limits_{k - 1}^{K}{a_{k}2^{- \frac{B_{k}}{M - 1}}}}}{{s.t.\mspace{14mu} {\sum\limits_{k = 1}^{K}B_{k}}} \leq B_{tot}}} & \left\lbrack {{equation}\mspace{14mu} 35} \right\rbrack\end{matrix}$

Here, we define a_(k) in modified CSI exchange method (method 2) asbelow equation.

$\begin{matrix}{a_{k} = \left\{ \begin{matrix}{a_{1} = {{\overset{\_}{\Gamma}(M)} \cdot {P\left( {{d_{K - 11}^{- \alpha}\lambda_{{{ma}\; x}\;}^{\lbrack{K - 12}\rbrack}} + {d_{K - 11}^{- \alpha}\lambda_{{ma}\; x}^{\lbrack{K - 11}\rbrack}}} \right)}}} \\{a_{2} = {{\overset{\_}{\Gamma}(M)} \cdot {P\left( {{d_{K\; 2}^{- \alpha}\lambda_{{ma}\; x}^{\lbrack{K\; 2}\rbrack}} + {d_{K - 11}^{- \alpha}\lambda_{{ma}\; x}^{\lbrack{K - 12}\rbrack}}} \right)}}} \\{{a_{k} = {{{{\overset{\_}{\Gamma}(M)} \cdot {Pd}_{k - {2k}}^{- \alpha}}\lambda_{{ma}\; x}^{\lbrack{k - {2k}}\rbrack}{\forall k}} = 3}},\ldots \mspace{14mu},K}\end{matrix} \right.} & \left\lbrack {{equation}\mspace{14mu} 36} \right\rbrack\end{matrix}$

And, a_(k) in star feedback method as below equation.

$\begin{matrix}{a_{k} = \left\{ {{{\begin{matrix}{a_{1} = {{\overset{\_}{\Gamma}(M)} \cdot {P\left( {{d_{K\; 2}^{- \alpha}\lambda_{{ma}\; x}^{\lbrack{K\; 1}\rbrack}} + {d_{K\; - 11}^{- \alpha}\lambda_{{ma}\; x}^{\lbrack{K - 11}\rbrack}}} \right)}}} \\{a_{2} = {{\overset{\_}{\Gamma}(M)} \cdot {P\left( {{d_{13}^{- \alpha}\lambda_{{ma}\; x}^{\lbrack 12\rbrack}} + {d_{K\; 2}^{- \alpha}\lambda_{{ma}\; x}^{\lbrack{K\; 2}\rbrack}}} \right)}}} \\{a_{k} = {{\overset{\_}{\Gamma}(M)} \cdot {P\left( {{d_{k - {1k} + 1}^{- \alpha}\lambda_{{ma}\; x}^{\lbrack{k - {1k}}\rbrack}} + {d_{k - {2k}}^{{- \alpha}|}\lambda^{\lbrack{k - {2k}}\rbrack}}} \right)}}} \\{{a_{K} = {{\overset{\_}{\Gamma}(M)} \cdot {P\left( {{d_{K - 11}^{- \alpha}\lambda_{{ma}\; x}^{\lbrack{K - {1K}}\rbrack}} + {d_{K - {2K}}^{- \alpha}\lambda_{{ma}\; x}^{\lbrack{K - 2}\rbrack}}} \right)}}},}\end{matrix}k} = 3},\ldots \mspace{14mu},{K - 1}} \right.} & \left\lbrack {{equation}\mspace{14mu} 37} \right\rbrack\end{matrix}$

Consider the objective function in equation 35. By forming thelagrangian and taking derivative with respect to B_(k), it can beexpressed by below equation.

$\begin{matrix}{{L = {{\sum\limits_{k \in U}{a_{k}2^{- \begin{matrix}B_{k} \\{M - 1}\end{matrix}}}} + {v\left( {{\sum\limits_{k \in U}B_{k}} - B_{tot}} \right)}}}{\frac{\partial L}{\partial B_{k}} = {{{{- 2^{- \begin{matrix}B_{k} \\{M - 1}\end{matrix}}}\ln \; 2\; \frac{a_{k}}{M - 1}} + v} = 0}}} & \left\lbrack {{equation}\mspace{14mu} 38} \right\rbrack\end{matrix}$

In equation 38, v is the Lagrange multiplier and U is the set offeedback link U={1, . . . K}. Therefore, we obtain B_(k) as belowequation.

$\begin{matrix}{B_{k} = {\left( {M - 1} \right) \cdot {\log_{2}\left( \frac{\mu \; a_{k}}{M - 1} \right)}}} & \left\lbrack {{equation}\mspace{14mu} 39} \right\rbrack\end{matrix}$

and B_(k) satisfies the following constraint equation where

$\begin{matrix}{{\mu = \frac{\ln \; 2}{v}}{{\sum\limits_{k \in \upsilon}{\left( {M - 1} \right) \cdot {\log_{2}\left( \frac{\mu \; a_{k}}{M - 1} \right)}}} = {B_{tot}.}}} & \left\lbrack {{equation}\mspace{14mu} 40} \right\rbrack\end{matrix}$

Combining equation 39 and 40 with B_(k)≧0, the number of optimalfeedback bit is obtained as below equation.

$\begin{matrix}{{B_{k}^{*} = {\frac{1}{U}\left( {\gamma - {\left( {M - 1} \right) \cdot {U} \cdot {\log_{2}\left( \frac{M - 1}{a_{k}} \right)}}} \right)^{\prime}}}{\mu = \left( {2^{B_{tot}}{\prod\limits_{k \in U}\left( \frac{M - 1}{a_{k}} \right)^{M - 1}}} \right)^{\frac{1}{{({M - 1})} \cdot {U}}}}{{In}\mspace{14mu} {equation}\mspace{14mu} 41},{\gamma = {B_{tot} + {\sum\limits_{k \in U}{\left( {M - 1} \right) \cdot {\log_{2}\left( \frac{M - 1}{a_{k}} \right)}}}}},} & \left\lbrack {{equation}\mspace{14mu} 41} \right\rbrack\end{matrix}$

|U| is the cardinality of U and

$(a)^{+} = \left\{ \begin{matrix}a & {{{if}\mspace{14mu} a} \geq 0} \\0 & {{{if}\mspace{14mu} a} < 0.}\end{matrix} \right.$

The solution of equation 41 is found iteratively through thewaterfilling algorithm, which is described as bellow.

Waterfilling algorithm.   i=0; U={1, . . . , K}; while i = 0 doDetermine the water-level$\gamma - B_{tot} + {\sum\limits_{k \in U}^{\;}{\left( {M - 1} \right) \cdot {\log_{2}\left( \frac{M - 1}{a_{k}} \right)}}}$. Choose the user set$\overset{\_}{k} = {\arg \; \max \left\{ {\frac{M - 1}{a_{k}}:{k \in U}} \right\}}$. if${\gamma - {\left( {M - 1} \right) \cdot {U} \cdot {\log_{2}\left( \frac{M - 1}{a_{k}} \right)}}} \geq 0$then optimal bit allocation B_(k)* in U is determined by equation 43.i=i+1; else Let define U = {U except for k} and B_(k)* = 0 .

From the waterfilling algorithm, we obtain {B_(k)*: kεU}. However,B_(k)* should become integer so that it is determined the optimal bitallocation B _(k)* as below equation.

B _(k) *=└B _(k)*┘

In equation 42, └x┘ is the largest integer not greater than x.

B. Scaling Law of Total Feedback Bits

In perfect CSI assumption, each pair of transmitter-receiver linkobtains the interference-free link for its desired data stream. However,misaligned beamformers due to the quantization error destroy the linearscaling gain of sum capacity at high SNR regime. In this section, weanalyze the total number of feedback bits that achieve a full networkDoF K in the proposed CSI feedback methods. The required sum feedbackbits are formulated as the function of the channel gains, the number ofantennas and SNR that maintain the constant sum rate loss over the wholeSNR regimes.

As P goes to infinity, the network DoF in K user interference channelachieves K with constant value of

$\sum\limits_{k - 1}^{K}{\log_{2}\left( {\hat{I}}^{\lbrack k\rbrack} \right)}$

according to below equation.

$\begin{matrix}\begin{matrix}{{DoF} = {\lim\limits_{P->\infty}\frac{R_{sum}^{limited}}{\log_{2}P}}} \\{= {\lim\limits_{P->\infty}\frac{\begin{matrix}{{\sum\limits_{k = 1}^{K}{\log_{2}\left( {\frac{P}{d_{kk}^{a}}{{{\hat{r}}^{\lbrack k\rbrack}H^{\lbrack{kk}\rbrack}{\hat{v}}^{\lbrack k\rbrack}}}^{2}} \right)}} -} \\{\sum\limits_{i = 1}^{K}{\log_{2}\left( {\hat{I}}^{\lbrack k\rbrack} \right)}}\end{matrix}}{\log_{2}P}}} \\{= {{\lim\limits_{P->\infty}\frac{\sum\limits_{k = 1}^{K}{\log_{2}\left( {d_{\underset{kk}{a}}^{P}{{{\hat{r}}^{\lbrack k\rbrack}H^{\lbrack{kk}\rbrack}{\hat{v}}^{\lbrack k\rbrack}}}^{2}} \right)}}{\log_{2}P}} -}} \\{{\lim\limits_{P->\infty}\frac{\sum\limits_{k = 1}^{K}{\log_{2}\left( {\hat{I}}^{\lbrack k\rbrack} \right)}}{\log_{2}P}}} \\{= {K - {\lim\limits_{P->\infty}\frac{\log_{2}\left( {\prod\limits_{k = 1}^{K}{\hat{I}}^{\lbrack k\rbrack}} \right)}{\log_{2}P}} - K}}\end{matrix} & \left\lbrack {{equation}\mspace{14mu} 43} \right\rbrack\end{matrix}$

Therefore, it can be formulated the sum residual interference as anexponential function of B_(k) as below equation.

$\begin{matrix}\begin{matrix}{{\sum\limits_{k = 1}^{K}{\log_{2}\left( {\hat{I}}^{\lbrack k\rbrack} \right)}} = {\log_{2}\left( {\prod\limits_{k = 1}^{K}{\hat{I}}^{\lbrack k\rbrack}} \right)}} \\{= {\log_{2}\left( {\prod\limits_{k = 1}^{K}{a_{k}2^{- \frac{B_{k}}{M - 1}}}} \right)}} \\{= c}\end{matrix} & \left\lbrack {{equation}\mspace{14mu} 44} \right\rbrack\end{matrix}$

Equation 44 yields an equation 45.

$\begin{matrix}\begin{matrix}{B_{tot}^{*} = {\sum\limits_{k = 1}^{K}B_{k}}} \\{= {\left( {M - 1} \right) \cdot \left( {{\sum\limits_{k}^{K}{\log_{2}a_{k}}} - {\log_{2}c}} \right)}} \\{= {{{K \cdot \left( {M - 1} \right) \cdot \log_{2}}P} + {\left( {M - 1} \right) \cdot}}} \\{\left( {{\sum\limits_{k = 1}^{K}{\log_{2}{\hat{a}}_{k}}} - c} \right)}\end{matrix} & \left\lbrack {{equation}\mspace{14mu} 45} \right\rbrack\end{matrix}$

In equation 45, c is constant and c>0 and a_(k)=P·â_(k), ∀k.

Since B_(tot)* is the integer number, we obtain the required feedbackbits B _(tot)* for achieving full DoF as below equation.

B _(tot)*=nint(B _(tot)*)  [equation 46]

In equation 46, nint(x) is the nearest integer function of x.

C. Implementation of Feedback Bits Controller

As we derived in equation 41 and 45, the computation of { B _(k)*}_(k−1)^(K) and B _(tot)* requires the set of channel gain {a_(k)}_(k=1) ^(K)that consist of the path-loss and short-term fading gain from allreceivers. Therefore, the optimal bit allocation strategy requires thecentralized bit controller that gathers full knowledge of {a_(k)}_(k=1)^(K) and computes the optimal set of feedback bits. The centralized bitcontroller can be feasible in star feedback method since it has a CSI-BSconnected with all receivers through the backhaul links.

However, the receiver in CSI exchange method only feeds back CSI tocorresponding transmitter, implemented by distributedfeedforward/feedback channel links. Therefore, the additional bitcontroller is required to allocate optimal feedback bits in CSI exchangemethod. Moreover, {a_(k)}_(k=1) ^(K) includes the gain of short-termfading so that CSI of all receivers should be collected to centralcontroller over every transmission period, which requires a largeoverhead of CSI exchange and causes delay that results in significantinterference misalignment in fast channel variation environments.

1) Path-loss based bit allocation method: To reduce the burden offrequent exchange of channel gains for bit allocation, we average{a_(k)}_(k=1) ^(K) over the short-term fading {H^([ij])}_(i,j=1) ^(K).Consider

E[λ_(ma x)^([ij])] ≤ M²

and apply it to equation 36. E_(H)[a_(k)] in modified CSI exchangemethod is represented as below equation where the expected value of ak'sconsists of transmit power, the number of antennas and path-loss.

$\begin{matrix}{{E_{H}\left\lbrack a_{k} \right\rbrack} = \left\{ \begin{matrix}\begin{matrix}{{E_{H}\left\lbrack a_{1} \right\rbrack} = {{{\overset{\_}{\Gamma}(M)} \cdot 2}{P \cdot M^{2} \cdot d_{K - 11}^{- \alpha}}}} \\{{E_{H}\left\lbrack a_{2} \right\rbrack} = {{\overset{\_}{\Gamma}(M)} \cdot {PM}^{2} \cdot \left( {d_{K\; 2}^{- \alpha} + d_{K - 11}^{- \alpha}} \right)}}\end{matrix} \\{{{E_{H}\left\lbrack a_{k} \right\rbrack} = {{\overset{\_}{\Gamma}(M)} \cdot {PM}^{2} \cdot d_{k - {2k}}^{- \alpha}}},{{\forall k} = 3},\ldots \mspace{14mu},{K.}}\end{matrix} \right.} & \left\lbrack {{equation}\mspace{14mu} 47} \right\rbrack\end{matrix}$

Applying equation 47 to 41 and 45, the dynamic bit allocation scheme canbe implemented by the path-loss exchange between receivers. Since thepath-loss shows a long-term variability compared with fast fadingchannel, the additional overhead for bit allocation is required over themuch longer period than the bit allocation scheme in equation 41 and 45.

2) Distributed bit allocation scheme in CSI exchange method: We developthe distributed bit allocation scheme achieving DoF K in modified CSIexchange method, without centralized bit controller. For the distributedbit allocation, we replace the condition (equation 44) that satisfiesDoF K as below equation.

$\begin{matrix}\begin{matrix}{{\log_{2}\left( {\hat{I}}^{\lbrack k\rbrack} \right)} = {\log_{2}\left( {a_{k}2^{- \frac{B_{k}}{M - 1}}} \right)}} \\{{= \frac{c}{K}},{{\forall k} = 1},\ldots \mspace{14mu},K}\end{matrix} & \left\lbrack {{equation}\mspace{14mu} 48} \right\rbrack\end{matrix}$

Therefore, the required feedback bit Bk for k-th beamformer is derivedas below equation where a_(k) is represented in equation 36.

$\begin{matrix}{B_{k} - {{\left( {M - 1} \right) \cdot \log_{2}}a_{k}} - {{\left( {M - 1} \right) \cdot \log_{2}}\frac{c}{K}}} & \left\lbrack {{equation}\mspace{14mu} 49} \right\rbrack\end{matrix}$

Combining equation 49 with 36, each receiver can compute the requiredfeedback bits with its own channel knowledge except for receiver K.Receiver K requires additional knowledge of d_(K−11) ^(−α)λ_(max)^([K−12]) from receiver K−1 to calculate a₂. From equation 49, it isprovided the design of modified CSI exchange method with bit allocationof achieving IA in distributed manners as below.

1. step 1; set-up: Receiver K forwards the matrix(H^([K2]))⁻¹H^([K1] to receiver K−)1.

2. step 2; computation of v^([1]): Receiver K−1 computes v^([1]) as anyeigenvector of (H^([K−1]))⁻¹H^([K−12])(H^([K2]))⁻¹H^([K1]) and quantizesit to {dot over (v)}^([1]) with B₁ computed by equation 49. {dot over(v)}^([1]) is forwarded to transmitter 1. Also, receiver K−1 forwardsd_(K−11) ^(−α)λ_(max) ^([K−12]) to receiver K.

3. step 3; computation of v[2]: Transmitter 1 feeds forward {circumflexover (v)}^([1]) to receiver K. Then, receiver K calculates {circumflexover (v)}^([2]) and quantizes it to {dot over (v)}^([2]) with B₂computed by equation 49. {dot over (v)}^([2]) is fed back to transmitter2.

4. step 4; computation of v^([3]), . . . , v^([K])

For i=3: K, Transmitter i−1 forwards {tilde over (v)}^([i−1]) toreceiver i−2. Then, receiver i−2 calculates {dot over (v)}_([i]) andquantizes it to {circumflex over (v)}_([i]) with B, computed by equation49. {circumflex over (v)}_([i]) is fed back to transmitter i−1.

V. Successive CSI Exchange Method Based on Precoded RS

CSI exchange method in previous description can reduce the amount ofnetwork overhead for achieving IA. However, it basically assumes thefeedforward/feedback channel links to exchange pre-determinedbeamforming vectors.

Considering precoded RS in LTE (long term evolution), the CSI exchangemethod can be slightly modified for the exchange of precoding vector(i.e. beamformer or beamforming vector). However, it still effectivelyscales down CSI overhead compared with conventional feedback method.

Consider K=M+1 user interference channel, where each transmitter andreceiver are equipped with M antennas and DoF(d) for each user is d=1.Based on IA condition and precoding vector in equation 6 and equation 7,we firstly explain the procedure of CSI exchange method infeedforward/feedback framework for K=4. After that, we describe theprocedure of CSI exchange method based on precoded RS, in detail.

1. Successive CSI exchange in method 1 (e.g. K=4)

In K=4, IA condition is shown as below equation.

$\begin{matrix}{\begin{matrix}\begin{matrix}\begin{matrix}{{{span}\left( {H^{\lbrack 13\rbrack}V^{\lbrack 3\rbrack}} \right)} = {{{span}\left( {H^{\lbrack 14\rbrack}V^{\lbrack 4\rbrack}} \right)}\mspace{14mu} {At}\mspace{14mu} {receiver}\mspace{14mu} 1}} \\{{{span}\left( {H^{\lbrack 21\rbrack}V^{\lbrack 1\rbrack}} \right)} = {{{span}\left( {H^{\lbrack 24\rbrack}V^{\lbrack 4\rbrack}} \right)}\mspace{14mu} {At}\mspace{14mu} {receiver}\mspace{14mu} 2}}\end{matrix} \\{{{span}\left( {H^{\lbrack 31\rbrack}V^{\lbrack 1\rbrack}} \right)} = {{{span}\left( {H^{\lbrack 32\rbrack}V^{\lbrack 2\rbrack}} \right)}\mspace{14mu} {At}\mspace{14mu} {receiver}\mspace{14mu} 3}}\end{matrix} \\{{{span}\left( {H^{\lbrack 42\rbrack}V^{\lbrack 2\rbrack}} \right)} = {{{span}\left( {H^{\lbrack 43\rbrack}V^{\lbrack 3\rbrack}} \right)}\mspace{14mu} {At}\mspace{14mu} {receiver}\mspace{14mu} 4}}\end{matrix}->\begin{matrix}\begin{matrix}\begin{matrix}{{{span}\left( V^{\lbrack 1\rbrack} \right)} = {{span}\left( {\left( H^{\lbrack 31\rbrack} \right)^{- 1}H^{\lbrack 32\rbrack}V^{\lbrack 2\rbrack}} \right)}} \\{{{span}\left( V^{\lbrack 2\rbrack} \right)} = {{span}\left( {\left( H^{\lbrack 42\rbrack} \right)^{- 1}H^{\lbrack 43\rbrack}V^{\lbrack 3\rbrack}} \right)}}\end{matrix} \\{{{span}\left( V^{\lbrack 3\rbrack} \right)} = {{span}\left( {\left( H^{\lbrack 13\rbrack} \right)^{- 1}H^{\lbrack 13\rbrack}V^{\lbrack 4\rbrack}} \right)}}\end{matrix} \\{{{span}\left( V^{\lbrack 4\rbrack} \right)} = {{span}\left( {\left( H^{\lbrack 24\rbrack} \right)^{- 1}H^{\lbrack 21\rbrack}V^{\lbrack 1\rbrack}} \right)}}\end{matrix}} & \left\lbrack {{equation}\mspace{14mu} 50} \right\rbrack\end{matrix}$

Then, precoding vectors at transmitter 1, 2, 3 and 4 can be determinedas below equation.

V ^([1])=eigen vector of (H ^([31]))⁻¹ H ^([32])(H ^([42]))⁻¹ H^([43])(H ^([13]))⁻¹ H ^([14])(H ^([24]))⁻¹ H ^([21])

V ^([2])=(H ^([32]))⁻¹ H ^([31]) V ^([1])

V ^([3])=(H ^([43]))⁻¹ H ^([42]) V ^([2])

V ^([4])=(H ^([14]))⁻¹ H ^([13]) V ^([3])

In equation 50, the interference from transmitter 3 and 4 are on thesame subspace at receiver 1. Likewise, the interference from transmitter1 and 4, the interference from transmitter 1 and 2, and the interferencefrom transmitter 2 and 3 are on the same subspace at receiver 2, 3, and4.

To inform the receiver that which interferers are aligned on the samedimension, CSI exchange method firstly requires initial signaling. Thisinitial signaling is transmitted with a longer period than that ofprecoder update.

After the initial signaling indicated which interferers are aligned ateach receiver, the receiver 1, 2, 3, 4 feed back the selectedinterference channel matrices multiplied by the inverse of anotherinterference channel matrices, {(H^([31]))⁻¹H^([32]),(H^([42]))⁻¹H^([43]), (H^([13]))⁻¹H^([14]), (H^([24]))⁻¹H^([21])} totransmitter 1. Based on these product channel matrices, transmitter 1determines V^([1]) and forwards it to receiver 3. Then, receiver 3determines V^([2]) using the channel information of(H^([32]))⁻¹H^([31]), and feeds it back to transmitter 2. With thisiterative way, V^([3]) and V^([4]) are sequentially determined. Thisprocess is already described in previous part. However, it is slightlydifferent in the point that VP) is determined by transmitter 1 not areceiver 3.

FIG. 5 shows the procedure of CSI exchange method in case that K is 4.

Referring FIG. 5 (a), the procedure of CSI exchange method comprisesbelow steps.

step 1. Initial Signaling: Each transmitter (e.g. BS) reports whichinterferers are aligned on the same dimension to its receiver. (In K=4,two of four interferers are selected as the aligned interference.)

step 2. Determine initial vector V^([1]): After initialization, receiver1, 2, 3, 4 feed back the product channel matrices to transmitter 1, thentransmitter 1 can determine VIII.

step 3. Computation of V^([2]) V^([3]) and V^([4]): Transmitter 1 feedsforward V^([1]) to receiver 3 and receiver 3 calculates V^([2]). Afterthat, receiver 3 feeds back V^([2]) to transmitter 2. With the exchangeof pre-determined precoding vector, the remained precoding vectors aresequentially computed.

Referring FIG. 5 (b), the step 3 in the CSI exchange method is shown,where FF is feedfoward and FB is feedback links.

In a system using precoded RS such as a LTE system, each transmitter(e.g. BS) transmits the reference symbol with precoding vector.Therefore, the feedfoward signaling in successive CSI exchange can bemodified compared with above CSI exchange method.

FIG. 6 shows the procedure of CSI exchange method based on precoded RSin case that K is 4.

Assume that i-th receiver perfectly estimates the channel matrices{H^([1i]), H^([2i]), . . . , H^([Ki])}. Then, precoding vectors V^([1]),V^([2]), V^([3]), and V^([4]) for IA are designed by equation 51 in K=4.To compute the precoding vectors, the procedure of CSI exchange methodbased on precoded RS is suggested as below.

Referring FIG. 6 (a), the procedure of CSI exchange method based onprecoded RS comprises below steps.

step 1. Initial signaling: Each BS (transmitter) reports whichinterferers are aligned on same dimension to its receiver. (The index ofaligned interferers is forwarded to the receiver.)

step 2. Determine initial vector V^([1]): After initialization, receiver1, 2, 3, 4 feed back the product channel matrices to transmitter 1, thentransmitter 1 can determine V″. Here, the product channel matricescomprises (H^([31]))⁻¹H^([32]), (H^([42]))⁻¹H^([43]),(H^([13]))⁻¹H^([14]) and (H^([24]))⁻¹H^([21]).

step 3. computation V^([2]), V^([3]) and V^([4]).

referring FIG. 6( b), the process of the step 3 includes belowprocedures.

Transmitter 1: After determining its precoder V^([1]), transmitter 1sends the precoded RS to receiver 3.

Receiver 3: Receiver 3 estimates effective channel H_(eff)^([31])=H^([31])V^([1]). Based on equation 51, receiver 3 computesV^([2]) multiplying H_(eff) ^([31]) with (H^([32]))⁻¹. Then, receiver 3sends V^([2]) to transmitter 2 through feedback channel.

Transmitter 2: Transmitter 2 sends the precoded RS based on V^([2]) toreceiver 4.

Receiver 4: Receiver 4 estimates effective channel H_(eff)^([42])=H^([42])V^([2]) and it computes V^([3]) multiplying H_(eff)^([42]) with (H^([43]))⁻¹. Then, V^([3]) is reported to transmitter 3through feedback channel.

Transmitter 3: Transmitter sends the precoded RS based on V^([3]) toreceiver 1.

Receiver 1: Receiver 1 estimates effective channel H_(eff)^([13])=H^([13])V^([3]) and receiver 1 computes V^([4]) multiplyingH_(eff) ^([13]) with (H^([14]))⁻¹. Then, V^([4]) is reported totransmitter 4 through feedback channel.

VI. Interference Power Control.

Interference misalignment (IM) due to CSI quantization errors results inresidual interference at receivers. The interference power at aparticular receiver depends on the transmission power at interferers,their beamformers' IM angles, and the realizations of interferencechannels. Since all links are coupled, there is a need for regulatingthe transmission power of transmitters such that the number of linksmeeting their QoS requirements, namely not in outage, is maximized. Asuitable approach is for the receivers to send TPC signals tointerferers to limit their transmission power. However, designing TPCalgorithms based on this approach is challenging due to the dependenceof interference power on several factors as mentioned above and therequired knowledge of global CSI and QoS requirements.

The TPC signals sent by a receiver provide upper bounds on thetransmission power of interferers (i.e. transmitter). First the effectof IM is described and then the method for transmitting a TPC signal foran interference power control is described.

1. The Effect of IM.

The IM due to CSI quantization errors results in residual interferenceat receivers as explained by the following example.

Consider the 3-user interference channel where the symbol extension(number of subchannels) is (2n+1)=3. Given v^([2])=1_(3×1) and based onIA condition, the 3×1 beamforming vectors (v₁ ^([1]), v₂ ^([1]),v^([2]), v^([3])) satisfy the following constraints in equation 52.

H ^([12]) v ^([2]) =H ^([13]) v ^([3])

H ^([23]) v ^([3]) =H ^([21]) v ₁ ^([1])

H ^([32]) v ^([2]) =H ^([31]) v ₂ ^([1])  [equation 52]

Here, the beamforming vector refers to weights applied to signalstransmitted over different subchannels rather than antennas in the caseof a multi-antenna system.

Generalizing the above constraints to an arbitrary n and usingzero-forcing linear receivers are sufficient to guarantee the optimalityin terms of DOF, namely each user achieves the optimal number of DOFequal to ½ per symbol as n converges to infinity. For the currentexample of n=1, the 3×1 zero-forcing beamforming vectors, denoted as (f₁^([1]), f₂ ^([1]), f^([2]), f^([3])), satisfy the following constraintsequation.

f ₁ ^([1])εnull[H ^([11]) v ₂ ^([1]) ,H ^([12]) v ^([2])]

f ₂ ^([1])εnull[H ^([11]) v ₁ ^([1]) ,H ^([12]) v ^([2])]

f ^([2])εnull[H ^([21]) v ₁ ^([1]) ,H ^([21]) v ₂ ^([1])]

f ^([3])εnull[H ^([31]) v ₁ ^([1]) ,H ^([32]) v ^([2])]  [equation 53]

It follows that the receive signals after zero-forcing equalization areinterference-free and given as below equation.

{circumflex over (x)} ₁ ^([1])=(f ₁ ^([1]))^(†) H ^([11]) v ₁ ^([1]) x ₁^([1])+(f ₁ ^([1]))^(†) z ^([1])

{circumflex over (x)} ₂ ^([1])=(f ₂ ^([1]))^(†) H ^([11]) v ₂ ^([1]) x ₂^([1])+(f ₂ ^([1]))^(†) z ^([2])

{circumflex over (x)} ^([2])=(f ^([2]))^(†) H ^([22]) v ^([2]) x^([2])+(f ^([2]))^(†) z ^([2])

{circumflex over (x)} ^([3])=(f ^([3]))^(†) H ^([33]) v ^([3]) x^([3])+(f ^([3]))^(†) z ^([3])  [equation 54]

In equation 54, z are the samples of the AWGN processes.

Let ({circumflex over (v)}₁ ^([1]), {circumflex over (v)}₂ ^([1]),{circumflex over (v)}^([3])) denote the quantized version of beamformingvectors (v₁ ^([1]), v₂ ^([1]), v^([3])). Due to CSI quantization errors,the IA condition in equation 52 are no longer satisfied. This isexpressed in equation 55.

H ^([12]) v ^([2]) ≠H ^([13]) {circumflex over (v)} ^([3])

H ^([23]) {circumflex over (v)} ^([3]) ≠H ^([21]) {circumflex over (v)}₁ ^([1])

H ^([32]) v ^([2]) ≠H ^([31]) {circumflex over (v)} ₂ ^([1])  [equation55]

Consequently, nulling interference using zero-forcing beamforming vectoris infeasible since it requires below equation.

H ^([12]) v ^([2]) =H ^([13]) {circumflex over (v)} ^([3])

H ^([23]) {circumflex over (v)} ^([3]) =H ^([21]) {circumflex over (v)}₁ ^([1])

H ^([32]) v ^([2]) =H ^([31]) {circumflex over (v)} ₂ ^([1])  [equation56]

It follows that the receive signals after zero-forcing equalization aregiven as below equation.

{circumflex over (x)} ₁ ^([1])=(f ₁ ^(1]))^(†) H ^(11]) {circumflex over(v)} ₁ ^([1]) x ₁ ^([1])+(f ₁ ^([1]))^(†)(H ^([11]) {circumflex over(v)} ₂ ^([1]) x ₂ ^([1]) +H ^([12]) {circumflex over (v)} ^([2]) x^([2]) +H ^([13]) {circumflex over (v)} ^([3]) x ^([3]))+(f ₁^([1]))^(†) z ^([1])

{circumflex over (x)} ₂ ^([1])=(f ₂ ^(1]))^(†) H ^(11]) {circumflex over(v)} ₂ ^([1]) x ₂ ^([1])+(f ₁ ^([1]))^(†)(H ^([11]) {circumflex over(v)} ₁ ^([1]) x ₁ ^([1]) +H ^([12]) {circumflex over (v)} ^([2]) x^([2]) +H ^([13]) {circumflex over (v)} ^([3]) x ^([3]))+(f ₂^([1]))^(†) z ^([2])

{circumflex over (x)} ^([2])=(f ^(2]))^(†) H ^(22]) {circumflex over(v)} ^([2]) x ^([2])+(f ₁ ^([2]))^(†)(H ^([21]) {circumflex over (v)} ₁^([1]) x ^([1]) +H ^([21]) {circumflex over (v)} ₂ ^([1]) x ₂ ^([1]) +H^([23]) {circumflex over (v)} ^([3]) x ^([3]))+(f ^([2]))^(†) z ^([2])

{circumflex over (x)} ^([3])=(f ^(3]))^(†) H ^(33]) {circumflex over(v)} ^([3]) x ^([3])+(f ₁ ^([1]))^(†)(H ^([31]) {circumflex over (v)} ₁^([1]) x ^([1]) +H ^([31]) {circumflex over (v)} ₂ ^([1]) x ₂ ^([1]) +H^([32]) {circumflex over (v)} ^([2]) x ^([2]))+(f ^([3]))^(†) z^([3])  [equation 57]

In equation 57, (f₁ ^([1]))^(†)(H^([11]){circumflex over (v)}₂ ^([1])x₂^([1)+H^([12]){circumflex over (v)}^([2])x^([2])+H^([13]){circumflexover (v)}^([3])x^([3])), (f₁ ^([1]))^(†)(H^([11]){circumflex over (v)}₁^([1])x₁ ^([1)+H^([12]){circumflex over(v)}^([2])x^([2])+H^([13]){circumflex over (v)}^([3])x^([3])), (f₁^(2]))^(†)(H^([21]){circumflex over (v)}₁^(1])x^([1])+H^([21]){circumflex over (v)}₂ ^([1])x₂^([1])+H^([23]){circumflex over (v)}^([3)x^([3])), (f₁^([1]))^(†)(H^([31]){circumflex over (v)}₁^([1])x^([1])+H^([31]){circumflex over (v)}₂ ^([1])x₂^([1])+H^([32]){circumflex over (v)}^([2])x^([2])), are residualinterferences and z are the samples of the AWGN processes. The residualinterferences in the above received signals are nonzero since it isinfeasible to choose receiver beamforming vectors in the null spaces ofthe interference.

2. The method for transmitting a TPC signal for an interference powercontrol.

We propose the method for transmitting a TPC signal between interferersand receivers such that the users' QoS constraints are satisfied. First,to compute the TPC signals, the users' QoS constraints can be translatedinto constraints on the interference power that are linear functions ofthe interferers' transmission power as elaborated below.

FIG. 7 shows effect of interference misalignment due to quantization offeedback CSI.

The quantization errors in feedback CSI causes IM at receivers.Specifically, at each receiver, interference leaks into the datasubspace even if linear zero-forcing interference cancellation isapplied as illustrated in FIG. 7.

Referring FIG. 7, the length of an arrow for a particular interfererdepends on the product of its transmission power and the interferencechannel. As observed from FIG. 7, the power of each residualinterference depends on the transmission power of the correspondinginterferer, the misalignment angles of IA beamforming vectors(beamformer), and the interference channel power. Thus, the method fortransmitting a TPC signal is to generate TPC signals based on thefactors including the misalignment angels of IA beamformers and theinterference channel power.

Let the power of the residual interference from the interferer n to thereceiver m be denoted by the function P^([n])f(θ^([mn])) where Prepresent the transmission power of the transmitter n and θ^([mn])represent the IM angle for the CSI sent from the receiver m to thetransmitter n, and f(θ^([mn]))≦sin²(θ^([mn])). In this case, the SINR atthe receiver m can be written as below equation.

$\begin{matrix}{{\min\limits_{\{\theta^{k}\}}{\sum\limits_{k = 1}^{K}{B^{\lbrack k\rbrack}\left( \theta^{\lbrack k\rbrack} \right)}}}{{s.t.\mspace{14mu} {constraint}}\mspace{14mu} {in}\mspace{14mu} {equation}\mspace{14mu} 65\mspace{14mu} {\forall m}}} & \left\lbrack {{equation}\mspace{14mu} 58} \right\rbrack\end{matrix}$

In equation 58, G^([mn]) represents the gain of the data link aftertransmit and receive beamforming and σ² represents the AWGN noisevariance. The QoS requirement for the m^(th) user can be quantified byan outage threshold η^([m]) at the m^(th) receiver whereSINR^([m])≧η^([m]) is required for correct data decoding. This leads toa constraint on the interference power expressed in below equation.

$\begin{matrix}{{\min\limits_{\{ P^{\lbrack k\rbrack}\}}{\lim\limits_{T->\infty}{\frac{1}{T}{\sum\limits_{t = 1}^{T}{\sum\limits_{k = 1}^{K}{B^{\lbrack k\rbrack}\left( \theta_{t}^{\lbrack k\rbrack} \right)}}}}}}{{s.t.\mspace{14mu} {constraint}}\mspace{14mu} {in}\mspace{14mu} {equation}\mspace{14mu} 65\mspace{14mu} {\forall m}}} & \left\lbrack {{equation}\mspace{14mu} 59} \right\rbrack\end{matrix}$

The above constraints on the residual interference's power can be usedto compute TPC signals sent by the n^(th) receiver for limiting thetransmission power {P^([m])|m≠n}.

Second, the sequential TPC signal exchange between subsets ofinterferers and interfered receivers is necessary for maximize thenumber of users who meet their QoS requirements. It is important to notethat what matters to a receiver is the sum residual interference powerbut this receiver needs to control individual interference components byTPC signal. Consequently, a challenge in computing TPC signals is toresolve the dilemma for a receiver that applies too stringent TPC on aparticular interferer may cause its data link to be in outage and tooloose TPC may cause the interferer to have strong interference onanother receiver. Thus TPC exchange provides a decentralized mechanismfor receivers to handshake and reach an optimally control of thetransmission power of all transmitters. Such TPC signal exchange willexploit the method of CSI exchange methods discussed earlier.

VII. Binary Power Control for Interference Alignment with LimitedFeedback

Given finite-rate CSI exchange, quantization errors in CSI causeinterference misalignment (IM). To control the resultant interferences,Transmission Power Control (TPC) method is needed. However, the optimalsolution for TPC is a non-convex problem.

For the simpler implementation, Binary Power Control (BPC) method isproposed. BPC method has the advantages of simpler power controlalgorithms. The sum rate capacity performance of BPC method in 2 users,which it is approaching 99% of the capacity of continuous power control.We expand this into general K multi-antenna user and propose adistributed method for implementing binary power control indecentralized networks.

A binary power control (BPC) method is the method that decides thetransmit power of transmitter K within P_(k)={0, P_(max)}. That is, asystem performed by the BPC method is easily understood as turningon/off system. For the practical use of transmission power control, wepropose the BPC method using feedback and feedforward between atransmitter and a receiver.

Consider K=M+1 MIMO user interference channel, where each transmitterand receiver are equipped with M antennas and DoF for each user is d=1.Based on IA condition and precoding vector in equation 50 and 51, wefirstly explain error propagation caused by quantization error in CSIexchange method (refer FIG. 2).

1. Error propagation caused by quantization by CSI exchange method (K=4,as an example).

If we assume that any two interferers are designed on the same dimensionat each receiver, the resultant precoder (beamforming vector,beamformer) can be expressed in closed-form as below equation.

V ^([1])=eigen vector of (H ^([31]))⁻¹ H ^([32])(H ^([42]))⁻¹ H^([42])(H ^([13]))⁻¹ H ^([14])(H ^([24]))⁻¹ H ^([21])

V ^([2])=(H ^([32]))⁻¹ H ^([31]) V ^([1])

V ^([3])=(H ^([43]))⁻¹ H ^([42]) V ^([2])

V ^([4])=(H ^([14]))⁻¹ H ^([13]) V ^([3])  [equation 60]

In equation 60, each precoder is normalized as V^([i])=V^([i])/∥V^([i])∥for i=1, 2, 3, 4.

The CSI exchange method prevents quantized matrix error by vector formfeedback and feedforward. However, while stages of feedback, thereinevitably exists an error propagation problem. In K=4, the beamformingvectors (precoding vectors) with quantization error is shown as belowequation.

{circumflex over (V)} ^([1]) =V ^([1]) +e ^([1])

{circumflex over (V)} ^([2]) =V ^([2]) +e ^([1]) +e ^([2])

{circumflex over (V)} ^([3]) =V ^([3]) +e ^([1]) +e ^([2]) +e ^([3])

{circumflex over (V)} ^([4]) =V ^([4]) +e ^([1]) +e ^([2]) +e ^([3]) +e^([4])  [equation 61]

In equation 61, e^([1]) is the channel quantization error frominitialization of V^([1]), e^([2]) to e^([4]) is the vector quantizationerror due to the feedback, and {circumflex over (V)}^([k]) is thebeamforming vector with the quantized channel matrices. Due to the errorpropagation, quantized vector errors are accumulated. Therefore,transmitter 4 causes biggest quantization error and this brings theresult of the probability of turning off for each transmitter as FIG. 8.

FIG. 8 shows probability of turning off for each transmitter.

Referring FIG. 8, probability of turning off is increased as the indexof transmitter is increased due to the error propagation.

Using this result, assuming that the receivers has the knowledge of allthe interfering channels, the BPC method start decision of the power ofuser K with the ratio γ_(k) expressed as following equation.

P _(k)=0 if γ_(k)<σ_(k) ² where γ_(k)=Σ_(i=1,i≠k) ^(K) P _(i) U ^([k])^(†) H ^([ki]) V ^([i])  [equation 62]

In equation 62, U^([k]) is the received M×1 beamforming vector forreceiver k, V[i] is the received M×1 beamforming vector for transmitteri, and H^([ki]) is the M×M channel matrix from transmitter i to receiverk, σ_(k) ² represents the AWGN noise variance.

2. A Binary Power Control (BPC) Method.

FIG. 9 shows a binary power control method.

Referring FIG. 9, the binary power control method comprises followingsteps.

1. step 1—Transmitter K: Receiver K computes γ_(k) using equation 62 andturns off the link (i.e P_(K)=0) if γ_(K)<σ_(K) ² and set the powerP_(K)=P_(Max) if γ_(K)≧σ_(K) ².

Then, the receiver K gives the feedback T^([K]) to transmitter K.T^([K]) denotes an indicator bit which indicates on/off. For example,T^([k])=1 if P_(k)=P_(Max) and T^([k])=0 if P_(k)=0. The power for allthe other transmitter is set as P_(Max).

2. step 2—Transmitter K−1: Receiver K−1 computes γ_(K−1) and turns offthe link if γ_(K−1)<σ_(K−1) ² and set the power P_(K−1)=P_(Max) ifγ_(K−1)≧σ_(K−1) ². Then, the receiver K−1 gives the feedback T^([K−1])to transmitter K−1. The power of transmitter K is set in the earlierstep (step 1) and T^([K]) is feedforwarded to receiver K−1. The powerfor all the other transmitter is considered as P_(Max). This process isrepeated until transmitter 2.

3. step 3—Transmitter 1: Receiver 1 computes γ₁ and decides its powerP₁={0, P_(Max)}. The power through transmitter 2 to K is set in earliersteps and feedforwarded to receiver 1

The described BPC method is using feedback and feedforward starting fromK^(th) user. Due to the error propagation, the K^(th) user has thebiggest quantization error. And for this reason, the K^(th) transmitterwith the most error happens to transmit with power ‘0’ with the biggestprobability. Using this result, this BPC method start decision of thepower of K^(th) transmitter for P_(k)={0, P_(Max)} with the ratio γ_(k)at the K^(th) receiver since the entire receiver has the knowledge ofall the interfering channels. This BPC method need not a central unit todetermine the network optimal on/off policy and the power of eachtransmitter is determined as distributed manner.

The solution for continuous power control is a non-convex problem.Therefore, the performance of continuous power control is obtained byexhaustive search. Exhaustive search is a method of generating randomsamples of combination of transmitters with its power uniformly between0 and P_(Max) and selects the combination that maximized the sum ratecapacity. The performance performed by the continuous power controlconverges to optimal result when the numbers of generated samples areenough.

To compare the sum rate capacity performance, binary exhaustive searchis used. To observe the sum rate capacity performance, first generateall (2^(K)−1) combinations of transmitters with its power as 0 orP_(Max), and select the combination that maximized the sum ratecapacity. In the case of K=4, M=3, the sample of combinations generatedare as following. (P_(Max), P_(Max), P_(Max), P_(Max)), (P_(Max),P_(Max), P_(Max), 0), . . . , (0, P_(Max), P_(Max), P_(Max)), (0, 0,P_(Max), P_(Max)), . . . , (0, 0, 0, P_(Max)).

FIG. 10 shows the result of the binary power control method.

Referring FIG. 10, When K=4, M=3, feedback bits=6 (using randomcodebook), the binary power control method approaches to 93% of thecapacity of exhaustive search when SNR is 20 dB, which is nearlyoptimal. The ideal curve is the result in perfect CSI environment.

3. Continuous Power Control Method.

The BPC method described above needs another timeslot for thefeedback/feedforward procedure for the power control. Also, the BPCmethod may incur a situation that too many links are turned off at highSNR environment. To reduce time delay and prevent the situation,continuous power control (CPC) method is proposed.

FIG. 11 shows a CPC method.

Referring FIG. 11, a CPC method comprises below procedures.

1. step 1—Receiver K−1: Receiver K−1 receives all the channelinformation from other receivers and defines transmitter's beamformingvector WI and transmits its quantized vector {circumflex over (V)}^([1])to transmitter 1.

2. step 2—Receiver K: Receiver K receives feedforward of {circumflexover (V)}^([1]) from transmitter 1 and defines beamforming vector V¹²).Then Receiver K calculates the residual interference with {circumflexover (V)}^([1]) and {circumflex over (V)}^([2]) which is the quantizedvector of V^([2]). Then, receiver K sets P₂, the power of transmitter 2under the constraint P₂ sin²θ_(K)≦Γ_(k). Γ_(k) is an interferencethreshold value of receiver k. The receiver K sends feedback data of{circumflex over (V)}^([2]) and P₂ to transmitter 2.

3. step 3—Receiver 1: Receiver 1 receives feedforward of P₂, {circumflexover (V)}^([2]) from transmitter 2 and defines beamforming vector Vu).Then Receiver 1 calculates the residual interference with and{circumflex over (V)}^([2]) which is the quantized vector of WI. Then,receiver 1 sets P₃, the power of transmitter 3 under the constraint P₃sin²θ₁≦Γ₁. Then receiver 1 sends feedback data of {circumflex over(V)}^([2]) and P₃ to transmitter 3. This process (i.e. step 3) isrepeated until receiver K−2.

4. step 4—Receiver K−1: After one cycle of feedback/feedforward,Receiver K−1 receives feedforward of PK, {circumflex over (V)}^([K])from transmitter K. And receiver K−1 sets P₁, the power of transmitter 1under the constraint P₁ sin²θ_(K−1)Γ_(K−1). The receiver K−1 sendsfeedback data of P₁ to transmitter 1.

The BPC method described above sends feedback and feedforward of theturning on/off signal(T^([K])) after setting beamforming vectors, but inthe CPC method, the power of each transmitter is set at the receiveralong with the decision of the beamforming vectors. The receiver decidesthe power of each transmitter with the interference threshold Γ_(k),which is predetermined parameter. The receiver k−2 compares the residualinterference P_(k) sin²θ_(k−2) with the threshold (Γ_(k-2)) and sets thepower of transmitter k.

While the invention has been described in connection with what ispresently considered to be practical exemplary embodiments, it is to beunderstood that the invention is not limited to the disclosedembodiments, but, on the contrary, is intended to cover variousmodifications and equivalent arrangements included within the spirit andscope of the appended claims.

1. A power control method for interference alignment in wireless networkhaving K transmitters and K receivers, the method comprising: receiving,performed by receiver n (n is an integer, 1≦n≦K−1), a power indicationsignal of transmitter n+1 from the transmitter n+1; determining,performed by the receiver n, power of transmitter n; and transmitting,performed by the receiver n, a power indication signal of transmitter nto the transmitter n, wherein the power of transmitter n is determinedbased on a residual interference of the receiver n, and the powerindication signal of transmitter n indicates a minimum transmissionpower or a maximum transmission power of transmitter n.
 2. The method ofclaim 1, further comprising: determining, performed by the receiver K,power of transmitter K based on a residual interference of the receiverK; and transmitting, performed by the receiver K, a power indicationsignal of transmitter K to the transmitter K.
 3. The method of claim 1,the power indication signal of transmitter n includes 1 bit whichindicates the minimum transmission power or the maximum transmissionpower of transmitter n.
 4. The method of claim 1, wherein K is 4 andeach of the K transmitters and the K receivers has 3 antennas.
 5. Themethod of claim 1, wherein the residual interference of the receiver nis determined based on the transmission power of transmitter i (i isinteger, 1≦i≦K, not n).
 6. A power control method for interferencealignment in wireless network having K transmitters and K receivers, themethod comprising: receiving, performed by a receiver n (n is aninteger, 1≦n≦K−2), information for transmission power and quantizedprecoding vector of transmitter n+1 from transmitter n+1; determining,performed by the receiver n, a precoding vector of a transmitter n+2;calculating, performed by the receiver n, a residual interference basedon the quantized precoding vector of transmitter n+1 and a quantizedprecoding vector of transmitter n+2, and a transmission power of thetransmitter n+2; and transmitting, performed by the receiver n, thequantized precoding vector of transmitter n+2 and information for thetransmission power of the transmitter n+2 to the transmitter n+2.
 7. Themethod of claim 6, further comprising: receiving, performed by areceiver K−1, channel information from other receivers; determining,performed by the receiver K−1, a quantized precoding vector of atransmitter 1 based on the channel information; and transmitting,performed by the receiver K−1, the quantized precoding vector of thetransmitter 1 to the transmitter
 1. 8. The method of claim 7, furthercomprising: receiving, performed by a receiver K, information forquantized precoding vector of transmitter 1 from transmitter 1;determining, performed by the receiver K, a precoding vector of atransmitter 2; calculating, performed by the receiver K, a residualinterference based on the quantized precoding vector of transmitter 1and a quantized precoding vector of transmitter 2, and a transmissionpower of the transmitter 2; and transmitting, performed by the receiverK, the quantized precoding vector of transmitter 2 and information forthe transmission power of the transmitter 2 to the transmitter
 2. 9. Themethod of claim 6, further comprising: receiving, performed by areceiver K−1, information for transmission power and quantized precodingvector of transmitter K from transmitter K; determining, performed bythe receiver K−1, a transmission power of a transmitter 1; andtransmitting, performed by the receiver K−1, the transmission power ofthe transmitter 1 to the transmitter
 1. 10. The method of claim 6,wherein K is 4 and each of the K transmitters and the K receivers has 3antennas.